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A005881
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Theta series of planar hexagonal lattice (A2) with respect to edge.
(Formerly M0187)
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2
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2, 2, 0, 4, 2, 0, 4, 0, 0, 4, 4, 0, 2, 2, 0, 4, 0, 0, 4, 4, 0, 4, 0, 0, 6, 0, 0, 0, 4, 0, 4, 4, 0, 4, 0, 0, 4, 2, 0, 4, 2, 0, 0, 0, 0, 8, 4, 0, 4, 0, 0, 4, 0, 0, 4, 4, 0, 0, 4, 0, 2, 0, 0, 4, 4, 0, 8, 0, 0, 4, 0, 0, 0, 6, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0, 6, 4, 0, 4, 0, 0, 4, 4, 0, 0, 4, 0, 4, 0, 0, 4, 4, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Also number of ways of writing n as the sum of a triangular number and three times a triangular number.
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
Given g.f. A(x), then q^(1/2)*A(q) is denoted phi_1(z) where q=exp(pi*i*z) in Conway and Sloane.
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REFERENCES
| J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 103. see Equ. (13)
M. D. Hirschhorn, The number of representations of a number by various forms, Discr. Math., 298 (2005), 205-211.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
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LINKS
| G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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FORMULA
| Expansion of q^(-1)*(a(q)-a(q^4))/3 in powers of q^2 where a() is a cubic AGM analog function. - Michael Somos Nov 05 2006
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MAPLE
| d:=proc(r, m, n) local i, t1; t1:=0; for i from 1 to n do if n mod i = 0 and i-r mod m = 0 then t1:=t1+1; fi; od: t1; end; [seq(2*(d(1, 3, 2*n+1)-d(2, 3, 2*n+1)), n=0..120)];
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PROG
| (PARI) {a(n)=if(n<0, 0, n=2*n+1; 2*sumdiv(n, d, kronecker(-12, d)*(n/d%2)))} /* Michael Somos Nov 05 2006 */
(PARI) {a(n)=if(n<0, 0, n=8*n+4; 2*sum(j=1, sqrtint(n\3), (j%2)*issquare(n-3*j^2)))} /* Michael Somos Nov 05 2006 */
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CROSSREFS
| a(n) = 2*A033762(n).
Sequence in context: A060821 A191718 A201396 * A144458 A098268 A128585
Adjacent sequences: A005878 A005879 A005880 * A005882 A005883 A005884
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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